The **Moore-Penrose pseudoinverse** of a matrix is a way of coming up with something like an inverse for a matrix that doesn’t have an inverse. If a matrix does have an inverse, then the pseudoinverse is in fact the inverse. The Moore-Penrose pseudoinverse is also called a generalized inverse for this reason: it’s not just *like* an inverse, it actually *is* an inverse when that’s possible.

Given an *m* by *n* matrix *A*, the Moore-Penrose pseudoinverse *A*^{+} is the unique *n* by *m* matrix satisfying four conditions:

*A**A*^{+}*A*=*A**A*^{+}*A**A*^{+}=*A*^{+}- (
*A**A*^{+})* =*A**A*^{+} - (
*A*^{+}*A*)* =*A*^{+}*A*

The first equation says that *A**A*^{+} is a left inverse for *A*, and *A*^{+}*A* is a right inverse for *A*.

The second equation says *A*^{+}*A* is a left inverse for *A*^{+}, and *A* *A*^{+} is a right inverse for *A*^{+}.

The third and fourth equations say that *A* *A*^{+} and *A*^{+}*A* are Hermitian.

If *A* is invertible, *A* *A*^{+} and *A*^{+}*A* are both the identity matrix. Otherwise *A* *A*^{+} and *A*^{+}*A* act an awful lot like the identity, as much as you could expect, maybe a little more than you’d expect.

## Galois connections and adjoints

John Baez recently wrote that a **Galois connection**, a kind of categorical **adjunction**, is

“the best approximation to reversing a computation that can’t be reversed.”

That sounds like a pseudoinverse! And the first two equations defining a pseudoinverse look a lot like things you’ll see in the context of adjunctions, so the pseudoinverse must be an adjunction, right?

The question was raised on MathOverflow and Michal R. Przybylek answered

I do not think the concept of Moore-Penrose Inverse and the concept of categorical adjunction have much in common (except they

both try to generalise the concept of inverse) …

and gives several reasons why. (Emphasis added.)

Too bad. It would have made a good connection. Applied mathematicians are likely to be familiar with Moore-Penrose pseudoinverses but not categorical adjoints. And pure mathematicians, depending on their interests, may be more familiar with adjoint functors than matrix pseudoinverses.

So what about John Baez’ comment? His comment was expository (and very helpful) but not meant to be rigorous. To make it rigorous you’d have to be rigorous about what you mean by “best approximation” etc. And when you define your terms carefully, in the language of category theory, you get adjoints. This means that the Moore-Penrose inverse, despite its many nice properties, doesn’t mesh well with categorical definitions.

Przybylek concludes

… adjunctions compose … but Moore-Penrose pseudoinverses—generally—do not. … pseudoinverses are not stable under

isomorphisms, thus are notcategorical.

This turns out to be an interesting example, but not of what I first expected. Rather than the pseudoinverse of a matrix being an example of an adjoint, it is an example of something that despite having convenient properties [1] does not compose well from a categorical perspective.

**Related posts**:

- What do you mean by “can’t”?
- How to differentiate a non-differentiable function
- Approximating a solution that doesn’t exist

[1] The book Matrix Mathematics devotes about 40 pages to stating theorems about the Moore-Penrose pseudoinverse.